{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:A33SDCWMRJCXF3KQTSLEUEKJGE","short_pith_number":"pith:A33SDCWM","schema_version":"1.0","canonical_sha256":"06f7218acc8a4572ed509c964a11493121680195c43d2a69638963f7a7ad0223","source":{"kind":"arxiv","id":"1509.08101","version":2},"attestation_state":"computed","paper":{"title":"Representation Benefits of Deep Feedforward Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky","submitted_at":"2015-09-27T15:26:58Z","abstract_excerpt":"This note provides a family of classification problems, indexed by a positive integer $k$, where all shallow networks with fewer than exponentially (in $k$) many nodes exhibit error at least $1/6$, whereas a deep network with 2 nodes in each of $2k$ layers achieves zero error, as does a recurrent network with 3 distinct nodes iterated $k$ times. The proof is elementary, and the networks are standard feedforward networks with ReLU (Rectified Linear Unit) nonlinearities."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1509.08101","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-27T15:26:58Z","cross_cats_sorted":["cs.NE"],"title_canon_sha256":"e8f7a721d0e2d8b4efb84ffddd21221dc7e3f65a67d26f672b3ffb42b0d82ccc","abstract_canon_sha256":"8c1540b682f662ebc9baf6bd2d163196aea3bf85cf1796ee676814f8a97c42d3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:52.569014Z","signature_b64":"5WDwCl2tzcdhSjbA/xmqKkpEzJZUX0Vog4rMPm7fpA7Yq0e1Vg79PvK/mGYaqjozWosfIp9CE5muBK7YRHjjDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06f7218acc8a4572ed509c964a11493121680195c43d2a69638963f7a7ad0223","last_reissued_at":"2026-05-18T01:31:52.568575Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:52.568575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Representation Benefits of Deep Feedforward Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NE"],"primary_cat":"cs.LG","authors_text":"Matus Telgarsky","submitted_at":"2015-09-27T15:26:58Z","abstract_excerpt":"This note provides a family of classification problems, indexed by a positive integer $k$, where all shallow networks with fewer than exponentially (in $k$) many nodes exhibit error at least $1/6$, whereas a deep network with 2 nodes in each of $2k$ layers achieves zero error, as does a recurrent network with 3 distinct nodes iterated $k$ times. The proof is elementary, and the networks are standard feedforward networks with ReLU (Rectified Linear Unit) nonlinearities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08101","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1509.08101","created_at":"2026-05-18T01:31:52.568645+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.08101v2","created_at":"2026-05-18T01:31:52.568645+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.08101","created_at":"2026-05-18T01:31:52.568645+00:00"},{"alias_kind":"pith_short_12","alias_value":"A33SDCWMRJCX","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_16","alias_value":"A33SDCWMRJCXF3KQ","created_at":"2026-05-18T12:29:10.953037+00:00"},{"alias_kind":"pith_short_8","alias_value":"A33SDCWM","created_at":"2026-05-18T12:29:10.953037+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":3,"internal_anchor_count":2,"sample":[{"citing_arxiv_id":"2605.21451","citing_title":"Approximation Theory for Neural Networks: Old and New","ref_index":55,"is_internal_anchor":true},{"citing_arxiv_id":"2401.01335","citing_title":"Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models","ref_index":189,"is_internal_anchor":true},{"citing_arxiv_id":"2604.05929","citing_title":"ReLU Networks for Exact Generation of Similar Graphs","ref_index":26,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE","json":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE.json","graph_json":"https://pith.science/api/pith-number/A33SDCWMRJCXF3KQTSLEUEKJGE/graph.json","events_json":"https://pith.science/api/pith-number/A33SDCWMRJCXF3KQTSLEUEKJGE/events.json","paper":"https://pith.science/paper/A33SDCWM"},"agent_actions":{"view_html":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE","download_json":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE.json","view_paper":"https://pith.science/paper/A33SDCWM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.08101&json=true","fetch_graph":"https://pith.science/api/pith-number/A33SDCWMRJCXF3KQTSLEUEKJGE/graph.json","fetch_events":"https://pith.science/api/pith-number/A33SDCWMRJCXF3KQTSLEUEKJGE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE/action/storage_attestation","attest_author":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE/action/author_attestation","sign_citation":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE/action/citation_signature","submit_replication":"https://pith.science/pith/A33SDCWMRJCXF3KQTSLEUEKJGE/action/replication_record"}},"created_at":"2026-05-18T01:31:52.568645+00:00","updated_at":"2026-05-18T01:31:52.568645+00:00"}